In this article, the dynamical behaviors of two classes of chaotic systems are considered based on generalized Lyapunov function theorem with integral inequalities. Explicit estimations of the ultimate bounds are derived. The results presented in this article contain the existing results as special cases. Computer simulation results show that the proposed method is effective. © 2014 Wiley Periodicals, Inc. Complexity 21: 363–369, 2015.

This paper is expected to introduce a novel memductor-based chaotic system. The local dynamical entities, such as the basic dynamical behavior, the divergence, the stability of equilibrium set, and the Lyapunov exponent, are all investigated analytically and numerically to reveal the dynamic characteristics of the new memductor-based chaotic system as the system parameters and the initial state of memristor change. Subsequently, an active control method is derived to study the synchronous stability of the novel memductor-based chaotic system (...) through making the synchronization error system asymptotically stable at the origin. Further to these, a memductor-based chaotic circuit is designed, realized, and applied to construct a new memductor-based secure communication circuit by employing the basic electronic components and memristor. Furthermore, the design principle of the memductor-based chaotic circuit is thoroughly analyzed and the concept of “the memductor-based chaotic circuit defect quantification index” is proposed for the first time to verify whether the chaotic output is consistent with the mathematical model. A good qualitative agreement is shown between the simulations and the experimental validation results. (shrink)

For synchronization of a class of chaotic systems in the presence of nonvanishing uncertainties, a novel time-varying gain observer-based sliding mode control is proposed. First, a novel time-varying gain disturbance observer is developed to estimate the uncertainties. Then, by using the output of TVGDO to modify sliding mode control, a new TVGDO-based SMC scheme is developed. Although the observation and control precision of conventional fixed gain disturbance observer-based control for chaotic systems can be guaranteed by a (...) high observer gain, the undesirable spike problem may be caused by the high gain if the initial values of estimate and true states are not equal. The most attractive feature of this work is that the newly proposed TVGDO can eliminate the spike problem by developing a time-varying gain scheme. Finally, the effectiveness of the proposed method is demonstrated by the numerical simulation. (shrink)

An amplitude controllable hyperjerk system is constructed for chaos producing by introducing a nonlinear factor of memristor. In this case, the amplitude control is realized from a single coefficient in the memristor. The hyperjerk system has a line of equilibria and also shows extreme multistability indicated by the initial value-associated bifurcation diagram. FPGA-based circuit realization is also given for physical verification. Finally, the proposed memristive hyperjerk system is successfully predicted with artificial neural networks for AI based engineering applications.

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new (...) system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system. (shrink)

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz-like chaotic system, which is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also (...) obtained for all the positive parameters values a,b,c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters ; that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization. (shrink)

This article introduces a simple and effective new algorithm for image encryption using a chaotic system which is based on the magic squares. This novel 3D chaotic system is invoked to generate a random key to encrypt any color image. A number of chaotic keys equal to the size of the image are generated by this chaotic system and arranged into a matrix then divided into non-overlapped submatrices. The image to be encrypted is also divided into (...) sub-images, and each sub-image is multiplied by a magic matrix to produce another set of matrices. The XOR operation is then used on the resultant two sets of matrices to produce the encrypted image. The strength of the encryption method is tested in two folds. The first fold is the security analysis which includes key space analysis and sensitivity analysis. In the second fold, statistical analysis was performed, which includes the correlation coefficients, information entropy, the histogram, and analysis of differential attacks. Finally, the time of encryption and decryption was computed and show very good results. (shrink)

In this paper, a 3D jerk chaotic system with hidden attractor was explored, and the dissipativity, equilibrium, and stability of this system were investigated. The attractor types, Lyapunov exponents, and Poincare section of the system under different parameters were analyzed. Additionally, a circuit was carried out, and a good similarity between the circuit experimental results and the theoretical analysis testifies the feasibility and practicality of the original system. Furthermore, a robust feedback controller was designed based on the finite-time stability (...) theory, which guarantees the synchronization of 3D jerk master-slave system in finite time and asymptotically converges to the origin. Finally, we also give verification for the discussion in this paper by numerical simulation. (shrink)

This paper investigates the aperiodic sampled-data control for a chaotic system. Firstly, Takagi–Sugeno fuzzy models for the chaotic systems are established. The lower and upper bounds of the sampling period are taken into consideration. Then, the criteria for mean square exponential stability analysis and aperiodic sampled-data controller synthesis are provided by means of linear matrix inequalities. And the real sampling patterns can be fully captured by constructing suitable Lyapunov functions. Finally, an illustrative example shows that the proposed (...) method is effective to guarantee that the system’s states are stable with aperiodic sampled data. (shrink)

A novel conservative chaotic system with no equilibrium is investigated in this study. Various dynamics such as the conservativeness, coexistence, symmetry, and invariance are presented. Furthermore, a partial-state feedback control scheme is proposed, and the stable domain of control parameters is analyzed based on the degenerate Hopf bifurcation. In order to verify the numerical simulation analysis, an analog circuit is designed. The simulation results show that the output of the analog circuit system can reproduce the numerical simulation results and (...) verify the correctness of the theoretical analysis. (shrink)

This paper proposes a new 5D chaotic system with the flux-controlled memristor. The dynamics analysis of the new system can also demonstrate the hyperchaotic characteristics. The design and analysis of adaptive synchronization for the new memristor-based chaotic system and its slave system are carried out. Furthermore, the modularized circuit designs method is used in the new chaotic system circuit implementation. The Multisim simulation and the physical experiments are conducted, compared, and matched with each other which can demonstrate (...) the existence of the attractor for the new system. (shrink)

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system (...) and yield a chaotic phase diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system. (shrink)

Chaotic systems with hidden multiscroll attractors have received much attention in recent years. However, most parts of hidden multiscroll attractors previously reported were repeated by the same type of attractor, and the composite of different types of attractors appeared rarely. In this paper, a memristor-based chaotic system, which can generate composite attractors with one up to six scrolls, is proposed. These composite attractors have different forms, similar to the Chua’s double scroll and jerk double scroll. Through theoretical (...) analysis, we find that the new system has no fixed point; that is to say, all of the composite multiscroll attractors are hidden attractors. Additionally, some complicated dynamic behaviors including various hidden coexisting attractors, extreme multistability, and transient transition are explored. Moreover, hardware circuit using discrete components is implemented, and its experimental results supported the numerical simulations results. (shrink)

The fault ride-through capability and fault current issues are the main challenges in doubly fed induction generator- based wind turbines. Application of the bridge-type fault current limiter was recognized as a promising solution to cope with these challenges. This paper proposes a nonlinear sliding mode controller for the BFCL to enhance the FRT performance of the DFIG-based WT. This controller has robust performance in unpredicted voltage sag level and nonlinear features. Theoretical discussions, power circuit, and nonlinear control consideration of the (...) SMC-based BFCL are conducted, and then, its performance is verified through time-domain simulations in the PSCAD/EMTDC environment. To reduce the chattering phenomenon and decrease the reaching time, it used the exponential reaching law for designed SMC. Also, the SMC-based BFCL performance is compared with the conventional and PI controller-based BFCL for both symmetrical and asymmetrical short-circuit faults. Simulation results reveal that the SMC-based BFCL provides better performance compared with the conventional and PI controller-based BFCL to enhance the FRT. (shrink)

Exploring the amplitude modulation phenomenon of chaotic signal has become a subject of great concern in recent years. This paper mainly concentrates on the preliminary study on amplitude modulation principle of a chaotic system. First, two 3D chaotic systems with quadratic product terms are introduced for studying the amplitude modulation phenomenon of chaotic signal. It is found that the signal amplitude of the first system can be controlled by partial quadratic coefficient. But for the second (...) system, none of nonlinear coefficient can be employed to control the signal amplitude. Then, the amplitude modulation principle of chaotic system is preliminarily studied by exploring the intrinsic relationship between nonzero equilibrium point and phase space trajectory, and it is further validated by introducing unified parameter to the two 3D chaotic systems. As a necessary condition, the principle provides a feasible and simple method for constructing and analyzing an amplitude modulation chaotic system. (shrink)

This study presents the fundamental concepts and technical details of a U-model-based control system design framework, including U-model realisation from classic model sets, control system design procedures, and simulated showcase examples. Consequently, the framework provides readers with clear understandings and practical skills for further research expansion and applications. In contrast to the classic model-based design and model-free design methodologies, this model-independent design takes two parallel formations: it designs an invariant virtual controller with a specified closed-loop transfer function in a feedback (...) control loop and it determines the real controller output by resolving the inverse of the plant U-model. It should be noted that this U-control provides a universal control system design platform for many existing linear/nonlinear and polynomial/state-space models and it complements many existing design approaches. Simulation studies are used as examples to demonstrate the analytically developed formulations and guideline for potential applications. (shrink)

Exploring the dynamics feature of robust chaotic system is an attractive yet recent topic of interest. In this paper, we introduce a three-dimensional fractional-order chaotic system. The important finding by analysis is that the position of signalx3descends at the speed of 1/cas the parameterbincreases, and the signal amplitude ofx1,x2can be controlled by the parametermin terms of the power function with the index −1/2. What is more, the dynamics remains constant with the variation of parametersbandm. Consequently, this system can (...) provide rich encoding keys for chaotic communication. By considering the properties of amplitude and position modulation, the partial projective synchronization and partial phase synchronization are realized with linear control scheme. The distribution map of optimal synchronization region in the control-parameter space is charted by defining the power consumption of controller. Numerical simulations are executed to confirm the theoretical analysis. (shrink)

This paper presents the design, simulation, and experimental verification of the fractional-order multiscroll Lü chaotic system. We base them on op-amp-based approximations of fractional-order integrators and saturated series of nonlinear functions. The integrators are first-order active realizations tuned to reduce the inaccuracy of the frequency response. By an exponential curve fitting, we got a convenient design equation for realizing fractional-order integrators of orders from 0.1 to 0.95. The results include simulations in SPICE of the mathematical description and the electronic (...) implementation and experimental measurements that confirm them. Monte Carlo and sensitivity tests revealed a robust realization. Contrary to its passive counterparts, the suggested realizations significantly reduce design and implementation efforts by favoring resistors and capacitors with commercial values and reducing hardware requirements. (shrink)

Computational modeling plays an important role in prediction and optimization of real systems and processes. Models usually have some parameters which should be set up to the proper value. Therefore, parameter estimation is known as an important part of the modeling and system identification. It usually refers to the process of using sampled data to estimate the optimum values of parameters. The accuracy of model can be increased by adjusting its parameters to the optimum value which need a richer (...) dataset. One simple solution for having a richer dataset is increasing the amount of data, but that can be costly and time consuming. When using data from animals or people, it is especially important to have a proper plan. There are several available methods for parameter estimation in dynamical systems; however there are some basic differences in chaotic systems due to their sensitivity to initial condition. Accordingly, in this paper, a new cost function which is proper for chaotic systems is applied to the chaotic one-dimensional map. Then the efficiency of a newly introduced intelligent method experimental design in extracting proper data is investigated. The results show the success of the proposed method. (shrink)

Due to the complex behavior of a multiscroll chaotic system, it is a good candidate for the secure communications. In this paper, by adding an additional variable to the modified Lorenz-type system, a new chaotic system that includes only linear and piecewise items but can generate 4n + 4 scroll chaotic attractors via choosing the various values of natural number n is proposed. Its dynamics including bifurcation, multistability, and symmetric coexisting attractors, as well as various chaotic (...) and periodic behaviors, are analyzed by means of attraction basin, bifurcation diagram, dynamic map, phase portrait, Lyapunov exponent spectrum, and C0 complexity in detail. The mechanism of the occurrence for generating multiscroll chaotic attractors is presented. Finally, this multiscroll chaotic system is implemented by using the Altera Cyclone IV EP4CE10F17C8 FPGA. It is found that this FPGA-based design has an advantage of requiring less resources for 0% of the embedded multipliers and 0% of the PLLs of this FPGA are occupied. (shrink)

Fourth-order autonomous nonlinear differential equations can exhibit chaotic properties. In this study, we propose a family of fourth-order chaotic systems with infinite equilibrium points whose equilibria form closed curves of different shapes. First, the phase diagrams and Lyapunov exponents of the system family are simulated. The results show that the system family has complex phase diagrams and dynamic behaviors. Simulation analysis of the Poincarè mapping and bifurcation diagrams shows that the system has chaotic characteristics. The circuit (...) simulation model is constructed and simulated in Multisim. The circuit simulation results coincide with the numerical simulation results, which verifies the circuit feasibility of the system. Then, based on Lyapunov stability theory and the adaptive control method, the synchronous control of the system with infinite equilibria is designed. Numerical simulation results verify that the system synchronization with the adaptive control method is well. Finally, the synchronous drive system is used for image encryption, the response system is used for decryption, and color image encryption is realized by combining deoxyribonucleic acid coding and operating rules. Therefore, this study not only enriched the research on infinite equilibria chaotic systems but also further expanded secure communication technology by combining chaotic synchronization control and DNA coding in image encryption. (shrink)

This paper proposes an integral sliding mode control method of a class of uncertain chaotic systems with saturation inputs. Firstly, fuzzy logic system is used to estimate the unknown nonlinear function. Then, a disturbance observer is constructed to estimate a compound disturbance, which contains the external disturbance, the error of saturation input and control output, and the fuzzy estimation error. Subsequently, a proposed integral sliding mode controller can ensure that all signals of the closed-loop system are ultimately bounded, (...) and based on the dynamic system of the integral sliding mode variable itself, the ultimate bound of the tracking error can be estimated. Simulation results show that the proposed ISMC method is more effective than the traditional ISMC method. (shrink)

In this paper, two scaling matrices are used to research the synchronization of different dimensional chaotic systems with unknown parameters. Firstly, the definition of synchronization of chaotic systems with different dimensions is introduced. Secondly, based on Lyapunov stability theorem and adaptive control method, an adaptive feedback hybrid controller and parameter adaptive laws are designed to realize synchronization of uncertain chaotic systems with different dimensions. Finally, three numerical experiments are carried out to verify the effectiveness (...) of the proposed method. (shrink)

Traditional philosophical discourse draws a distinction between ontology and epistemology and generally enforces this distinction by keeping the two subject areas separated and unrelated. In addition, the relationship between the two areas is of central importance to physics and philosophy of physics. For instance, all kinds of measurement-related problems force us to consider both our knowledge of the states and observables of a system (epistemic perspective) and its states and observables independent of such knowledge (ontic perspective). This applies to quantum (...) systems in particular. (shrink)

This is the first work to apply complex systems science to the psychological interplay of order and chaos. The author draws on thought from a wide range of disciplines-both conventional and unorthodox-to address such questions as the nature of consciousness, the relation between mind and reality, and the justification of belief systems. The material should provoke thought among systems scientists, theoretical psychologists, artificial intelligence researchers, and philosophers.